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The Joint Density of the Surplus Before and After Ruin in the Sparre Andersen Model

Part of: Special processes Mathematical economics

Published online by Cambridge University Press:  14 July 2016

Susan M. Pitts  and
Konstadinos Politis
Susan M. Pitts*
Affiliation:
University of Cambridge
Konstadinos Politis*
Affiliation:
University of Piraeus
*
∗Postal address: Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK. Email address: [email protected]
∗∗Postal address: Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli and Dimitriou St., Piraeus 18534, Greece.
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Abstract

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Gerber and Shiu (1997) have studied the joint density of the time of ruin, the surplus immediately before ruin, and the deficit at ruin in the classical model of collective risk theory. More recently, their results have been generalised for risk models where the interarrival density for claims is nonexponential, but belongs to the Erlang family. Here we obtain generalisations of the Gerber-Shiu (1997) results that are valid in a general Sparre Andersen model, i.e. for any interclaim density. In particular, we obtain a generalisation of the key formula in that paper. Our results are made more concrete for the case where the distribution between claim arrivals is phase-type or the integrated tail distribution associated with the claim size distribution belongs to the class of subexponential distributions. Furthermore, we obtain conditions for finiteness of the joint moments of the surplus before ruin and the deficit at ruin in the Sparre Andersen model.

Keywords

Sparre Andersen modelsurplus prior to ruindeficit at ruinladder heightWiener-Hopf factorsphase-type distributionsubexponential distribution

MSC classification

Secondary: 60K10: Applications (reliability, demand theory, etc.) 91B30: Risk theory, insurance 60K05: Renewal theory
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 3 , September 2007 , pp. 695 - 712
Copyright
Copyright © Applied Probability Trust 2007 

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